Calculation of the aeolian sediment flux-density profile based on estimation of the kernel density

Aeolian Research 16 (2015) 49–54

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Calculation of the aeolian sediment flux-density profile based on estimation of the kernel density Meng Li ⇑, Zhibao Dong, Zhengcai Zhang Key Laboratory of Desert and Desertification, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, No. 322, West Donggang Road, Lanzhou, Gansu Province 730000, China

a r t i c l e

i n f o

Article history: Received 19 June 2014 Revised 11 November 2014 Accepted 11 November 2014

Keywords: Sediment flux-density profile Nonparametric method Kernel density estimation Bandwidth

a b s t r a c t Aeolian sediment flux is an important issue of aeolian research. Parametric estimation is a traditional method in which aeolian sediment flux is estimated based on parameterization of a chosen equation. This method is simple, but has some limitations; specifically, it requires a priori assumptions about the density distribution that may not be correct. In this study, we applied a popular and extensively used, data-driven, nonparametric method called kernel-density estimation to calculate the aeolian sediment flux-density profile. Nonparametric methods make no prior assumption about the form of the density distribution to be estimated; instead, the aim is to obtain an empirical estimate from the data that can provably converge on the true density that would be obtained using an infinite sample size. Through the calculation of aeolian sediment flux based on kernel-density estimation, we determined that the key point in this method is not selection of the kernel function, but rather the selection of the optimal bandwidth, which is a difficult task. The results of our calculations showed that the method is both computationally feasible and acceptably accurate. Equally significantly, the idea of applying nonparametric methods to the calculation of aeolian sediment fluxes may lead to the development of a suite of other related analytical and modeling methods. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Wind erosion of soil leads to ecosystem degradation and various hazards to human values in arid and semiarid areas, which make up one-third of the world’s surface (Lal, 1990; Sterk and Raats, 1996). This erosion damages valuable and nonrenewable soil resources, and the sediments generated in the process of erosion may form huge clouds that block sunlight, pollute water, damage crops and herds, and even threaten human life. In addition, climate and weather may be influenced by dust suspended in the atmosphere, since the dust reflects, scatters, diffuses, and absorbs solar radiation (Han et al., 2009). As human activities such as land reclamation and over-grazing can make the climate dryer and interact with any long-term warming and drying trends, the exposure of more soil to the wind exacerbates the problem of soil erosion (Dong et al., 2000). During the process of wind erosion, sediment particles are generated and transported by the wind in one of three modes (suspension, saltation, or creep), depending on the aerodynamic properties of the particles and the strength of the wind. Even within the same mode, particles vary in their speed, direction, ⇑ Corresponding author. Tel.: +86 931 496 7485. E-mail address: [email protected] (M. Li). http://dx.doi.org/10.1016/j.aeolia.2014.11.003 1875-9637/Ó 2014 Elsevier B.V. All rights reserved.

acceleration, and other motion parameters. Due to these variations, particles are dispersed to different heights above the ground and form a sediment cloud. Parametric estimation is a traditional method of aeolian sediment flux research that has been used to describe this cloud. In this approach, the aeolian sediment flux profile is assumed to be described by a mathematical function with several parameters. Some distribution functions have been widely adopted, such as the exponential and logarithmic distributions. After the distribution function has been chosen, its parameters are estimated according to the observed data. The literature on sediment flux research based on this approach includes data generated by wind-tunnel tests (Butterfield, 1999; Dong et al., 2006), field observations (Greeley et al., 1996; Namikas, 2003), and numerical simulations and theoretical analyses (Anderson and Haff, 1988, 1991; Zheng et al., 2004; Kang et al., 2008; Shi and Huang, 2010). Compared with parametric estimation methods, nonparametric methods make no prior assumption about the form of the flux density to be estimated. They are therefore both flexible and capable of reducing modeling biases, and can potentially generate more robust and accurate estimates. Their biggest advantage is that a supposed distribution function is not required a priori, thereby avoiding the problem of inadvertently selecting an inappropriate

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model. In addition, where outliers exist in the data, parametric methods may fail to capture the complete structure of the actual curve. Non-parametric estimation methods alleviate this problem by treating each observation as a part of the model. Non-parametric estimation methods include histogram estimation (Triola, 2010), Rosenblatt (1956) estimation, Parzen (1962) kernel-density estimation, and nearest-neighbor estimation (Wasserman, 2007). The histogram estimation method has been applied extensively because it is simple and intuitive, but the size range of the observed data must be known in advance, and the density estimation curve is discrete. For this reason, the Rosenblatt and Parzen kernel-density estimation methods were developed. Rosenblatt estimation does not require a subdivision strategy for the data and the intervals are calculated rather than assumed, so that data points always lie in the middle of the interval. It has been mathematically demonstrated that the estimator obtained is close to the true value (Rosenblatt, 1956). In Parzen kernel-density estimation, each estimated point has a fixed neighborhood. If the neighborhood size is large, dense data points exert excessive influence on the overall distribution, causing flattening of curves and potentially eliminating spikes that represent important information. In contrast, sparse points and outliers may be ignored because of their small neighborhood, and estimates for these neighborhoods may be zero even though a non-zero result would be more accurate or physically realistic. Loftsgaarden and Quesenberry (1965) developed nearest-neighbor estimation to mitigate the problems with Parzen kernel-density estimation. In nonparametric methods, it is necessary to account for the bandwidth, which represents a smoothing factor that is used to reduce the effect of spikes in the density distribution (i.e., to account for the effect of outliers), thereby producing a more regular function that does not completely ignore the effects of outliers. When the bandwidth is large, kernel-density estimation functions better than the nearest-neighbor method, which is not recommended, but many scholars nonetheless use this method to sort the data. Efron (1979, 1982) and Efron and Stein (1981) presented a nonparametric estimation method called bootstrapping, which produced a model that fit the actual distribution, but with a relatively high error. Silverman and Young (1987) decreased the mean squared error (MSE) of the bootstrapping method. Katkovnik and Shmulevich (2002) proposed a variable-window kernel-density estimation method which requires only the knowledge of the variance of the estimate. By means of numerical simulations, this method performed significantly better than any constant-bandwidth method. In this study, we examined the improved kernel-density estimation method developed by Parzen (1962) with the goal of identifying the key factors that affect the use of this method. We then applied the method to calculate the wind-blown sediment flux and compared the results with empirical data. 2. Kernel-density estimation Kernel-density estimation attempts to estimate an unknown density function based on probability theory. This method has existed for decades and some early discussions on kernel-density estimations can be found in Rosenblatt (1956) and in Parzen (1962). Ruppert and Cline (1994) proposed a modified kernel-density estimation based on a clustering algorithm for the dataset’s density function. As computers become more capable of handling high burden computation, research interests have increased.

K() is a probability-density function for the kernel and n is the sample size. The kernel-density estimation for fn(x) is defined as:

  n X x  Xi ^f n ðxÞ ¼ 1 ; K nh i¼1 h

8x 2 R

ð1Þ

where n represents the sample size, the positive constant h is called the bandwidth, i is the sample number, and R represents the set of real numbers. 2.2. Selection of kernel functions Kernel-density estimation deals with more than just obtaining an appropriate sample; it also requires careful estimation of the kernel function and the bandwidth. All three factors determine the performance of the estimation. Kernel functions must meet the following requirements:

Non-negativity : KðxÞ > 0;

8x 2 R

ð2Þ

Symmetry : KðxÞ ¼ KðxÞ;

8x 2 R

ð3Þ

Normalization :

Z

þ1

KðxÞ ¼ 1

ð4Þ

1

Commonly used kernel functions (Wasserman, 2007) include the triangular, Epanechnikov, quartic, triweight, Gaussian, cosine, and exponential functions. In this work, we found that some kernel functions, such as cosine kernel, Epanechnikov kernel and quartic kernel functions, were not appropriate for the calculation of the aeolian sediment flux-density profile, because they are confined as |(x  Xi)/h| 6 1. Only Gaussian kernel and exponential kernel are appropriate for this calculation. In practice, bandwidth selection becomes more important, as we will demonstrate in this paper. 2.3. Estimation of the bandwidth It is important to choose an appropriate bandwidth to provide an accurate estimation of the kernel’s density distribution. Ideally, the bandwidth should be as low as possible to avoid over-smoothing the curve, but high enough to remove spikes in the estimated distribution that would distort the description of the empirical data. In a univariate case, the performance of the kernel-density estimation depends strongly on the bandwidth, which functions as a weight function for the estimated kernel. Selection of the optimal bandwidth is a crucial problem in kernel-density estimation and has been the subject of considerable theoretical research, especially in the context of univariate kernel-density estimation (Dutta, 2011). These efforts include studies by Rudemo (1982), Bowman (1984), Silverman (1986), Scott and Terrell (1987), Park and Marron (1990), Jones and Kappenman (1991), Cao et al. (1994), Marron and Ruppert (1994), Wand and Jones (1995), and Simonoff (1996). The method for global and local bandwidth selection is the mean integrated square error (MISE) criterion (Wasserman, 2007). Familiarization with the MISE criterion is not required for the practical use of the kernel-density estimation, but it will help those who are interested, learn how one rigorously arrives to a well-chosen bandwidth.

Z  Z 2 2 ½^f ðxÞ  f ðxÞ dx ¼ E½^f ðxÞ  f ðxÞ dx MISE ¼ E

ð5Þ

2.1. The model definition In our study, we started with the model of Parsen method. First, draw a random sample X1, X2, . . ., Xn from the density function f(x).

where the density estimation ^f ðxÞ is a kernel-density estimation of f(x) and is a function of the bandwidth h, E represents the statistical expectation. Our approach of bandwidth estimation is to minimize

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the value of Eq. (7). As an unbiased estimation of MISE, the integrated square error (ISE) (Wasserman, 2007) is defined as:

ISE ¼

Z

2 ½^f ðxÞ  f ðxÞ dx

ð6Þ

Now our goal is to choose a value of h that minimizes the value of Eq. (9) instead of Eq. (7). This step is a complicated process (Wand and Jones, 1995; Mnatsakanov and Sarkisian, 2012), but the calculations can be performed using the MATLAB software (http://www.mathworks.com/products/matlab/).

smooth to reflect the characteristics of the distribution, and the bias of the estimator is large. In contrast, the influence of random disturbances creates spikes that cannot be ignored if the bandwidth (h1 = 0.75, Re1 = 0.57) is too small, and the estimator is not accurate. The best estimated curve (h2 = 0.95, Re2 = 0.53) provides a good fit and it is smooth enough. Therefore, besides the MISE criterion, we can also experimentally obtain the value of h through the smooth degree of the curves. Usually, the value of h can be obtained by cross validation or rules of thumb (Silverman, 1986), because the MISE method is sometimes too complicated though it is accurate.

3. Application and results 3.2. Probability-density curves under different kernel functions We used kernel-density estimation to calculate the aeolian sediment flux profile above a surface. To support this analysis, we collected field data on sediment flux at the Shapotou Aeolian Experiment Site (37°320 N, 105°020 E) in 2006 using the LDDSEG sampler (Dong et al., 2011), which is capable of trapping 80% of the blowing particles (Dong et al., 2012). Our goal was to describe the distribution of the sediment mass as a function of the height above the surface. Previous research had only studied the sand flux near the surface (Dong et al., 2002). In the present study, the sand sampler was 100 cm tall, and we extended the height in our models from the surface to 50 cm. 3.1. Probability-density curves under different bandwidths To clarify the importance of bandwidth selection, we compared the probability-density curves under different bandwidths using a Gaussian kernel:

 2 1 x ; KðxÞ ¼ pffiffiffiffiffiffi exp  2 2p

8x 2 R

Based on the actual conditions for a cloud of blowing sediment and for the sake of simplicity, we chose Gaussian and exponential kernels to show the effect of kernel choice. Eq. (12) is the probability-density function for a Gaussian kernel. The probability-density function for an exponential kernel can be written as:

   n X x  X i  ^f n ðxÞ ¼ 1   exp  nh i¼1 h 

ð9Þ

Fig. 2 shows that the Gaussian kernel (Re = 0.5258) produces a smoother curve and a better fit than the exponential kernel (Re = 1.4774). In theory (Wasserman, 2007), both curves could provide good robustness. We believe that the main reason why the exponential curve is not as good as the Gaussian curve is numerical calculation. Therefore, in this paper, Gaussian kernel was adopted as the kernel function. Also, more effort would be put on improving the algorithm for these two kernels in future research.

ð7Þ 3.3. Calculation of aeolian sediment flux

The associated probability-density function can be written as

" # n 1 X ðx  X i Þ2 ^f n ðxÞ ¼ pffiffiffiffiffiffi ; exp  2 2pnh i¼1 2h

8x 2 R

ð8Þ

Based on the range of values from previous research (Bashtannyk and Hyndman, 2001), we chose four bandwidths to illustrate the results (Fig. 1). For this work, evaluation of the goodness of the curve fitting employing correlation coefficient (r2) is not appropriate (Ellis et al., 2009), so root mean squared error (RMSE) was adopted in this paper. Inspection of Fig. 1 reveals that with the largest bandwidth, the estimated curve (h4 = 3.00, Re4 = 2.91) is too

Fig. 1. The probability-density of the curves for a Gaussian kernel under different bandwidths (h).

The total aeolian sediment flux can be calculated from the associated flux-density profile. We used the Gaussian kernel function (Eq. (12)) for this analysis because of its good mathematical properties (Triola, 2010), and Fig. 2 showed that this provided a good fit for the empirical data. The percentage of the total sediment mass flux in an arbitrary height interval (between heights a and b) of height can be achieved through integration:

Pða 6 x 6 bÞ ¼

Z a

b

n X 1 ^f n ðxÞdx ¼ pffiffiffiffiffiffi 2pnhn j¼1

Z a

b

" exp 

ðx  xj Þ2 2

2hn

# dx

ð10Þ

Fig. 2. Probability-density curves using two different kernel functions. For both curves, we used a bandwidth of h = 0.95.

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M. Li et al. / Aeolian Research 16 (2015) 49–54

where j is the number of height intervals. We defined the transformation y = (2x  a  b)/(b  a) to facilitate the programming and calculations: n ba X Pða 6 x 6 bÞ ¼ pffiffiffiffiffiffi 2pnhn j¼1

Z

1

( exp 

½ðb  aÞy þ a þ b  2xj 

1

2

8hn

2

) dy ð11Þ

Eq. (11) can be calculated using the Gauss–Legendre integral formula provided by Matlab. 3.4. Comparison with a parametric method There are many parametric methods for calculation of the aeolian sediment flux profile. We chose a frequently used parametric method (Ellis et al., 2009; Dong et al., 2011; Zhang et al., 2012), based on an exponential function, to provide a comparison with the kernel density method. This function can be written as:

qðzÞ ¼ cez=d

ð12Þ

where q is the sediment flux density, z is the height from the sand surface, and c and d are estimated regression parameters. This exponential function provides a simple and accurate estimation method, and in previous research, it provided as good a fit as a more complex function with more parameters (Dong et al., 2011). Experimental data were obtained at our study site from March to August 2006. In the kernel-density method, we used a bandwidth of ha = 0.95, hb = 0.97, hc = 0.95, hd = 0.85, he = 0.85, hf = 0.95. Fig. 3 shows that the kernel-density method (Rea = 0.49, Reb = 0.64, Rec = 0.58, Red = 0.24, Ree = 0.42, Ref = 0.53) provides results that are close to the parametric method (Rea = 0.40, Reb = 0.45, Rec = 0.51, Red = 0.72, Ree = 0.98, Ref = 0.44), and that the Gaussian kernel-density method can provide good fit for the observed data. In Fig. 3(d) and (e), kernel-density method produces a better fit than the parametric method. In Fig. 3(a) and (c), we can see two obvious outliers. We believe that the outliers are probably due to the experimental error. To a certain degree, the parametric method can avoid these outliers. This is one of the major advantages of the parametric method. In this non-parametric method, these outliers can be avoided through

Fig. 3. Comparison of the parametric (exponential function) and nonparametric (kernel density) methods for calculation of the aeolian sediment flux-density profiles based on data from March to August in 2006.

M. Li et al. / Aeolian Research 16 (2015) 49–54

enhancing the curve smooth degree. We are looking for the balance between smoothness and accuracy. In Fig. 3, of all the six kernel-density curves, only one reaches the bottom observed point. The reason is that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated (Karunamuni and Alberts, 2005). Some methods (Jones, 1993; Hall and Park, 2002) can alleviate this boundary problem but cannot completely solve it. Without regard to this bottom point, the kernel-density method provide a much better fitting of observed data than the parametric method. In Fig. 3(f), the kernel-density curve (Re = 0.16) fits better than the parametric curve (Re = 0.38) without calculating this bottom point. So the estimation of the bottom point needs to be improved. If this problem can be solved, this method would make more sense. Dong et al. (2011) found that a three-parameter modified exponential function provided the best fit for the sediment flux-density profiles in a desertified area of the Minqin area of northwestern China. The parametric function used to describe the aeolian sediment flux-density profiles may vary if the height range being investigated changes. However, because kernel-density estimation is specifically designed to estimate an unknown density function, it may have wider applicability because it makes no a priori assumption about the form of the density distribution that will be estimated. In addition, it is only used for data-driven density estimation and can therefore adapt to the characteristics of the dataset. In contrast, the parametric method has at least two important limitations: First, the prior choice of a distribution function determines the success of the estimation (i.e., the inherent degree of bias), no matter how much the model parameters are adjusted. Second, the equations that have been commonly used by previous researchers were monotonic, and could not account for the possibilities of fluctuations in the relationship.

4. Conclusions Though some parametric methods provide a good fit for aeolian sediment flux profiles, it is not possible to guarantee ab initio that a given model will be appropriate for a given situation. Moreover, they face the limitation of monotonicity. Nonparametric methods do not face these limitations, so at least in theory, they should be more stable and adaptable to a wider range of situations. The results of the present analyses, using empirical field data, demonstrate that kernel-density estimation is an effective and practical method of computation. The key to successful use of this method is not the selection of the best kernel function, since most functions that meet the constraints for using this method will produce similar results; rather, selection of an optimal bandwidth is crucial. The estimation of this bandwidth using the MISE method is complicated, so the method requires further improvement. The expression used to describe the kernel is more complicated than the equations used in most parametric methods, but it can be calculated easily enough by computer software and did not require the estimation of model parameters. As a new method to calculate the aeolian sediment flux-density profile, there are some problems with this nonparametric method inevitably. Although our results demonstrate that the kernel-density method provides results close to the true values, we must still improve our algorithm to confirm that the numerical transformation we chose does not affect the accuracy. How to avoid the effect of outliers and how to estimate the bottom point accurately will be important aspects in our future work on estimation of the kernel density. These results are significant because the use of nonparametric methods for calculating the aeolian sediment flux may lead to

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